Exponential loss of memory for the dynamic $\Phi^4_2$ with small noise

We consider the dynamic $\Phi^4_2$ model (or stochastic Allen--Cahn equation) formally given by the SPDE \begin{equation*} (\partial_t - \Delta) \phi_\varepsilon = - \phi_\varepsilon^3 + \phi_\varepsilon +\sqrt{2\varepsilon} \xi \end{equation*} where $\xi$ is a space-time white noise. When $\varepsilon$ is small the solutions of the equation spend long time intervals in metastable states before reaching equilibrium. This phenomenon is known as metastability.

We discuss a coupling argument for solutions started from suitable initial conditions in space dimension 2 as a consequence of metastability. Such a result is already known in space dimension 1. The basic obstacle in our case is that classical solution theory for SPDEs is not applicable here since the non-linear term is ill-defined due to the irregularity of $\xi$. Hence a renormalization is required to compensate the divergences of the non-linear term. This destroys the "nice" structure of the non-linear term and a deeper analysis on the level of the so-called "remainder" term is required. An interesting application of such a coupling argument appears in the proof of the Eyring--Kramers law in the theory of metastability.